NMAA13034U Introduction to K-theory
(K-Theory)

K-theory associates to a C^*-algebra A a couple of abelian
groups K_0(A) and K_1(A) that on one hand contain deep information
about the algebra A and on the other hand they can be calculated
for great many algebras. K-theory is one of the most important
constructions in both non-commutative geometry and in topology with
a host of applications in mathematics, and in physics. For
commutative unital C^*-algebras, alias continuous functions on
compact spaces, there are two equivalent descriptions of the
K-groups, each with its own advantages. In one description K_0
classifies (stable) projections and in the other description it
classifies (stable) vector bundles over the compact space(the
spectrum) associated to the algebra.The course will stress both
viewpoints.

The course will contain the following specific elements:

Projections in C^*-algebras and vector bundles

The Grassmannian and classification of vector bundles

The Grothendieck construction af K-theory

Exact sequences and calculation of K-groups.

K-theory of C_0(X) and Bott isomorphism

Products in K-theory.

The course is intended both for student in Non-commutative
Geometry and students in Topology.

Knowledge: The student will obtain knowledge of the elements
mentioned in the description of the content

Skills: After completing the course the student will be able to
1. calculate K-groups
2. classify projections in C^*-algebras and vector bundles
3. translate between the C^*-algebra and the vector bundle approach

Competences:
After completing the course the student will be able to
1. prove theorems within the subject of the course
2. apply the theory to both topology and non-commutative
geometry
3. understand the extensive litterature on elementary K-theory and
to read the more advanced parts of the subject.